Two Methods For Solving Nonlinear Ill-posed Problems

With the development of scientific and technology, the research of inverseproblem has been a popular subject which is across the computationalmathematics, applied mathematics and system science driven by the requirementof other fields of science and technology in many engineering applications. Manyfields in mathematics can almost put forward some form of inverse problems. Thestudy of inverse problem is closely related to the modernization of industry andnational defense and the application of science and technology in medicine andphysics. Most of inverse problem is ill-posed, there for, its effective solution andapplication are significant for mathematics and applied science, and sometimesare the crux of the problem. To sum up, the study of nonlinear ill-posed problemsare necessary and valuable.This paper is divided into four parts. First of all, the purpose andsignificance of the study of nonlinear ill-posed problems are introduced. Then thestatus of research at home and abroad is analyzed. Secondly, ill-posed nonlinearproblems are analyzed. This paper summarized several classical methods forsolving nonlinear ill-posed problem. Thirdly, the research on Landweber iterativemethod in the nonlinear ill-posed problem is carried on. In the condition of thenonlinear operator and the data approximation, a double circulation Landweberiterative dual perturbation method has been put forward based on FrozenLandweber iterative method, and it’s proved to be effective after the analysis ofthe monotonicity and convergence. Finally, according to the application ofTikhonov regularization method in the nonlinear ill-posed problems, a newiteration format is got with the Tikhonov regularization in the three-order ofmodified Newton method. And the iterative format is proved to be effective bychoosing the suitable regularization parameters, making use of the generalized error criterion under appropriate conditions and analyzing its monotonicity andconvergence.